Mathematics

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6 pages, 215 KiB

Open AccessArticle

The Use of the Evenness of Eigenvalues of Similarity Matrices to Test for Predictivity of Ecosystem Classifications

by Enrico Feoli and Paola Ganis

Mathematics 2019, 7(3), 245; https://doi.org/10.3390/math7030245 - 9 Mar 2019

Cited by 6 | Viewed by 2382

Abstract

The use of the evenness (E(λ)) of the eigenvalues of similarity matrices corresponding to different hierarchical levels of ecosystem classifications, is suggested to test correlation (or predictivity) between biological communities and environmental factors as one alternative of analysis of [...] Read more.

The use of the evenness (E(λ)) of the eigenvalues of similarity matrices corresponding to different hierarchical levels of ecosystem classifications, is suggested to test correlation (or predictivity) between biological communities and environmental factors as one alternative of analysis of variance (parametric or non-parametric). The advantage over traditional methods is the fact that similarity matrices can be obtained from any kind of data (mixed and missing data) by indices such as those of Goodall and Gower. The significance of E(λ) is calculated by permutation techniques. One example of application of E(λ) is given by a data set describing plant community types (beech forests of the Italian peninsula). Full article

(This article belongs to the Special Issue New Paradigms and Trends in Quantitative Ecology)

14 pages, 2577 KiB

Open AccessArticle

Aggregating a Plankton Food Web: Mathematical versus Biological Approaches

by Ferenc Jordán, Anett Endrédi, Wei-chung Liu and Domenico D’Alelio

Mathematics 2018, 6(12), 336; https://doi.org/10.3390/math6120336 - 19 Dec 2018

Cited by 14 | Viewed by 4187

Abstract

Species are embedded in a web of intricate trophic interactions. Understanding the functional role of species in food webs is of fundamental interests. This is related to food web position, so positional similarity may provide information about functional overlap. Defining and quantifying similar [...] Read more.

Species are embedded in a web of intricate trophic interactions. Understanding the functional role of species in food webs is of fundamental interests. This is related to food web position, so positional similarity may provide information about functional overlap. Defining and quantifying similar trophic functioning can be addressed in different ways. We consider two approaches. One is of mathematical nature involving network analysis where unique species can be defined as those whose topological position is very different to others in the same food web. A species is unique if it has very different connection pattern compared to others. The second approach is of biological nature, based on trait-based aggregations. Unique species are not easy to aggregate with others because their traits are not in common with the ones of most others. Our goal here is to illustrate how mathematics can provide an alternative perspective on species aggregation, and how this is related to its biological counterpart. We illustrate these approaches using a toy food web and a real food web and demonstrate the sensitive relationships between those approaches. The trait-based aggregation focusing on the trait values of size (sv) can be best predicted by the mathematical aggregation algorithms. Full article

(This article belongs to the Special Issue New Paradigms and Trends in Quantitative Ecology)

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12 pages, 419 KiB

Open AccessArticle

Statistical Analysis of Maximally Similar Sets in Ecological Research

by David W. Roberts

Mathematics 2018, 6(12), 317; https://doi.org/10.3390/math6120317 - 11 Dec 2018

Viewed by 2931

Abstract

Maximally similar sets (MSSs) are sets of elements that share a neighborhood in a high-dimensional space defined by a symmetric, reflexive similarity relation. Each element of the universe is employed as the kernel of a neighborhood of a given size (number of members), [...] Read more.

Maximally similar sets (MSSs) are sets of elements that share a neighborhood in a high-dimensional space defined by a symmetric, reflexive similarity relation. Each element of the universe is employed as the kernel of a neighborhood of a given size (number of members), and elements are added to the neighborhood in order of similarity to the current members of the set until the desired neighborhood size is achieved. The set of neighborhoods is then reduced to the set of unique, maximally similar sets by eliminating all sets that are permutations of an existing set. Subsequently, the within-MSS variability of candidate explanatory variables associated with the elements is compared to random sets of the same size to estimate the probability of obtaining variability as low as was observed. Explanatory variables can be compared for effect size by the rank order of within-MSS variability and random set variability, correcting for statistical power as necessary. The analyses performed identify constraints, as opposed to determinants, in the triangular distribution of pair-wise element similarity. In the example given here, the variability in spring temperature, summer temperature, and the growing degree days of forest vegetation sample units shows the greatest constraint on forest composition of a large set of candidate environmental variables. Full article

(This article belongs to the Special Issue New Paradigms and Trends in Quantitative Ecology)

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14 pages, 5828 KiB

Open AccessArticle

Evaluating the Predictive Power of Ordination Methods in Ecological Context

by Otto Wildi

Mathematics 2018, 6(12), 295; https://doi.org/10.3390/math6120295 - 1 Dec 2018

Cited by 6 | Viewed by 4637

Abstract

When striving for the ordination methods best predicting independently measured site factors, the following questions arise: does the optimal choice depend on the kind of biological community analysed? Are there different ordination methods needed to address different site factors? Simultaneously, I explore alternative [...] Read more.

When striving for the ordination methods best predicting independently measured site factors, the following questions arise: does the optimal choice depend on the kind of biological community analysed? Are there different ordination methods needed to address different site factors? Simultaneously, I explore alternative similarity approaches of entire ordinations, as well as the role of the transformations applied to the scale used in measuring species performance. The combination of methods and data transformations results in 96 alternative solutions for any one data set. These are compared by a graphical display, that is, an ordination of ordinations. The goodness-of-fit of independently measured site factors is assessed by two alternative methods. The resulting 96 performance values serve as independent variables in trend surfaces overlaid to the ordination of ordinations. The results from two real-world data sets indicate that some ordination methods greatly vary with data transformation. Scores close to a binary scale perform best in almost all ordination methods. Methods that intrinsically constrain the range of species scores, such as principal components analysis based on correlation, correspondence analysis (including its detrended version), nonmetric multidimensional scaling, as well as principal coordinates analysis based on the Bray-Curtis distance, always figure among the most successful methods, irrespective of data used. Full article

(This article belongs to the Special Issue New Paradigms and Trends in Quantitative Ecology)

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16 pages, 1161 KiB

Open AccessArticle

Identifying the Informational/Signal Dimension in Principal Component Analysis

by Sergio Camiz and Valério D. Pillar

Mathematics 2018, 6(11), 269; https://doi.org/10.3390/math6110269 - 20 Nov 2018

Cited by 13 | Viewed by 4071

Abstract

The identification of a reduced dimensional representation of the data is among the main issues of exploratory multidimensional data analysis and several solutions had been proposed in the literature according to the method. Principal Component Analysis (PCA) is the method that [...] Read more.

The identification of a reduced dimensional representation of the data is among the main issues of exploratory multidimensional data analysis and several solutions had been proposed in the literature according to the method. Principal Component Analysis (PCA) is the method that has received the largest attention thus far and several identification methods—the so-called stopping rules—have been proposed, giving very different results in practice, and some comparative study has been carried out. Some inconsistencies in the previous studies led us to try to fix the distinction between signal from noise in PCA—and its limits—and propose a new testing method. This consists in the production of simulated data according to a predefined eigenvalues structure, including zero-eigenvalues. From random populations built according to several such structures, reduced-size samples were extracted and to them different levels of random normal noise were added. This controlled introduction of noise allows a clear distinction between expected signal and noise, the latter relegated to the non-zero eigenvalues in the samples corresponding to zero ones in the population. With this new method, we tested the performance of ten different stopping rules. Of every method, for every structure and every noise, both power (the ability to correctly identify the expected dimension) and type-I error (the detection of a dimension composed only by noise) have been measured, by counting the relative frequencies in which the smallest non-zero eigenvalue in the population was recognized as signal in the samples and that in which the largest zero-eigenvalue was recognized as noise, respectively. This way, the behaviour of the examined methods is clear and their comparison/evaluation is possible. The reported results show that both the generalization of the Bartlett’s test by Rencher and the Bootstrap method by Pillar result much better than all others: both are accounted for reasonable power, decreasing with noise, and very good type-I error. Thus, more than the others, these methods deserve being adopted. Full article

(This article belongs to the Special Issue New Paradigms and Trends in Quantitative Ecology)

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17 pages, 1415 KiB

Open AccessArticle

A Generalized Framework for Analyzing Taxonomic, Phylogenetic, and Functional Community Structure Based on Presence–Absence Data

by János Podani, Sandrine Pavoine and Carlo Ricotta

Mathematics 2018, 6(11), 250; https://doi.org/10.3390/math6110250 - 12 Nov 2018

Cited by 11 | Viewed by 5136

Abstract

Community structure as summarized by presence–absence data is often evaluated via diversity measures by incorporating taxonomic, phylogenetic and functional information on the constituting species. Most commonly, various dissimilarity coefficients are used to express these aspects simultaneously such that the results are not comparable [...] Read more.

Community structure as summarized by presence–absence data is often evaluated via diversity measures by incorporating taxonomic, phylogenetic and functional information on the constituting species. Most commonly, various dissimilarity coefficients are used to express these aspects simultaneously such that the results are not comparable due to the lack of common conceptual basis behind index definitions. A new framework is needed which allows such comparisons, thus facilitating evaluation of the importance of the three sources of extra information in relation to conventional species-based representations. We define taxonomic, phylogenetic and functional beta diversity of species assemblages based on the generalized Jaccard dissimilarity index. This coefficient does not give equal weight to species, because traditional site dissimilarities are lowered by taking into account the taxonomic, phylogenetic or functional similarity of differential species in one site to the species in the other. These, together with the traditional, taxon- (species-) based beta diversity are decomposed into two additive fractions, one due to taxonomic, phylogenetic or functional excess and the other to replacement. In addition to numerical results, taxonomic, phylogenetic and functional community structure is visualized by 2D simplex or ternary plots. Redundancy with respect to taxon-based structure is expressed in terms of centroid distances between point clouds in these diagrams. The approach is illustrated by examples coming from vegetation surveys representing different ecological conditions. We found that beta diversity decreases in the following order: taxon-based, taxonomic (Linnaean), phylogenetic and functional. Therefore, we put forward the beta-redundancy hypothesis suggesting that this ordering may be most often the case in ecological communities, and discuss potential reasons and possible exceptions to this supposed rule. Whereas the pattern of change in diversity may be indicative of fundamental features of the particular community being studied, the effect of the choice of functional traits—a more or less subjective element of the framework—remains to be investigated. Full article

(This article belongs to the Special Issue New Paradigms and Trends in Quantitative Ecology)

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Review

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13 pages, 782 KiB

Open AccessReview

Scaling Laws in the Fine-Scale Structure of Range Margins

by Beáta Oborny

Mathematics 2018, 6(12), 315; https://doi.org/10.3390/math6120315 - 9 Dec 2018

Cited by 6 | Viewed by 2819

Abstract

Margins of the geographic distributions of species are important regions in terms of ecological and evolutionary processes, including the species’ response to climate change. This paper reviews some spatially explicit metapopulation models of range margins across environmental gradients (e.g., across latitudes or altitudes). [...] Read more.

Margins of the geographic distributions of species are important regions in terms of ecological and evolutionary processes, including the species’ response to climate change. This paper reviews some spatially explicit metapopulation models of range margins across environmental gradients (e.g., across latitudes or altitudes). These models share some robust results, which allow for generalizations within a broad variety of species and environments: (1) sharp edges can emerge even across relatively smooth environmental gradients; (2) intraspecific competition combined with dispersal limitation is a sufficient condition for the sharpening; (3) at the margin, the “mainland” of continuous occurrence splits into “islands”. Computer simulations pointed out some characteristic scaling laws in the size distribution of the islands, and in the structure of the hull of the mainland. The hull is a fractal with a dimension 7/4. Its width and length scale with the gradient according to characteristic scaling laws (with exponents 3/7 and 4/7, respectively). These general features follow from a second-order phase transition from a connected to a fragmented state. The results contribute to understanding the origin of vegetation zones and the spatial pattern of ecotones. Full article

(This article belongs to the Special Issue New Paradigms and Trends in Quantitative Ecology)

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Other

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16 pages, 855 KiB

Open AccessPerspective

Ecoacoustics: A Quantitative Approach to Investigate the Ecological Role of Environmental Sounds

by Almo Farina

Mathematics 2019, 7(1), 21; https://doi.org/10.3390/math7010021 - 26 Dec 2018

Cited by 60 | Viewed by 7494

Abstract

Ecoacoustics is a recent ecological discipline focusing on the ecological role of sounds. Sounds from the geophysical, biological, and anthropic environment represent important cues used by animals to navigate, communicate, and transform unknown environments in well-known habitats. Sounds are utilized to evaluate relevant [...] Read more.

Ecoacoustics is a recent ecological discipline focusing on the ecological role of sounds. Sounds from the geophysical, biological, and anthropic environment represent important cues used by animals to navigate, communicate, and transform unknown environments in well-known habitats. Sounds are utilized to evaluate relevant ecological parameters adopted as proxies for biodiversity, environmental health, and human wellbeing assessment due to the availability of autonomous audio recorders and of quantitative metrics. Ecoacoustics is an important ecological tool to establish an innovative biosemiotic narrative to ensure a strategic connection between nature and humanity, to help in-situ field and remote-sensing surveys, and to develop long-term monitoring programs. Acoustic entropy, acoustic richness, acoustic dissimilarity index, acoustic complexity indices (ACIt_f and ACIf_t and their evenness), normalized difference soundscape index, ecoacoustic event detection and identification routine, and their fractal structure are some of the most popular indices successfully applied in ecoacoustics. Ecoacoustics offers great opportunities to investigate ecological complexity across a full range of operational scales (from individual species to landscapes), but requires an implementation of its foundations and of quantitative metrics to ameliorate its competency on physical, biological, and anthropic sonic contexts. Full article

(This article belongs to the Special Issue New Paradigms and Trends in Quantitative Ecology)

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